If `a , b , c` are non-zero real numbers and if the system of equations `(a-1)x=y=z` `(b-1)y=z+x` `(c-1)z=x+y` has a non-trivial solution, then prove that `a b+b c+c a=a b c`

Explanation

Given equations `(a−1)x−y−z=0,`

`x−(b−1)y+z=0,`

`x+y−(c−1)z=0 `

can be written as AX=O

where `A=abs([​a−1,-1,-1],[​1,−(b−1),1],[​1,1,−(c−1)])​`

,`X=abs([​x],[y],[z])​`

,`O=abs([0],[0],[0])`


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